Estimating conditional hazard functions and densities with the highly-adaptive lasso

TL;DR

This paper introduces a highly-adaptive lasso method for estimating conditional hazard functions and densities, achieving optimal convergence rates even under data censoring and when traditional empirical risk minimizers are ill-defined.

Contribution

It develops a novel highly-adaptive lasso estimator that works effectively for complex, multivariate data with censoring, outperforming traditional methods in convergence and basis function efficiency.

Findings

The proposed estimator achieves optimal convergence rates under smoothness assumptions.

It provides a new conditional density estimator with proven convergence rate.

The method is effective even when empirical risk minimizers are ill-defined or inconsistent.

Abstract

We consider estimation of conditional hazard functions and densities over the class of multivariate c\`adl\`ag functions with uniformly bounded sectional variation norm when data are either fully observed or subject to right-censoring. We demonstrate that the empirical risk minimizer is either not well-defined or not consistent for estimation of conditional hazard functions and densities. Under a smoothness assumption about the data-generating distribution, a highly-adaptive lasso estimator based on a particular data-adaptive sieve achieves the same convergence rate as has been shown to hold for the empirical risk minimizer in settings where the latter is well-defined. We use this result to study a highly-adaptive lasso estimator of a conditional hazard function based on right-censored data. We also propose a new conditional density estimator and derive its convergence rate. Finally, we show that the result is of interest also for settings where the empirical risk minimizer is well-defined, because the highly-adaptive lasso depends on a much smaller number of basis function than the empirical risk minimizer.